Both feet are constrained by both positions.
Windup: pivot in contact with the rubber, non-pivot on or behind the front of the rubber extended.
Set: pivot either on or directly in front of and parallel to the rubber, non-pivot in front of the rubber extended.

A practical advantage might explain why pitchers adopt it, but such an advantage (which I certainly don't deny) would not provide a reason to prohibit it by rule. That's what we need to answer Tborze's question.

Yes, it's an illegal pitching position. My state wants this enforced in every game at every (HS) level. I teach a 2 step procedure:
98% of the time, we can address the problem in warmups: let coach know he's in the "hybrid" (in my state, all coaches have heard about this ad nauseam), and have coach address it (usually, "get back on the rubber, Bubba").
If he suddenly starts using the hybrid, address it when you see it. It's not illegal until the pitching motion begins (often a rocker step), at which time we kill it and award a ball to the batter.
We don't have to assume. The set position is clearly defined and requires the pivot foot to be (completely, entire foot) on the rubber OR in front of, touching, and parallel to the rubber. In either case, the entire foot (more or less) needs to be within the 24" length of the rubber. The hybrid does not satisfy either requirement. But yes, if he's not legal, first motion is an illegal pitch, and with runners on that's a balk.
I get this question all the time. The answer is: I can see his feet. If he's not clearly illegal, then he's legal. PU/BU is immaterial, no protractor required.
I've already addressed this, but since you asked, it's worth adding that the position is not illegal until he pitches from it. Being on the rubber in the hybrid is not in itself a violation. If he disengages and fixes it, then he's legal (no matter what the opposing coach says). And the time for me to fix it expires when the ball becomes live.
In theory, yes: the windup (let's say) yields a better pitch, and the set yields a better pick (in FED, the only legal pick). If F1 gets to have both, or the appearance of both, then that's an advantage. Why bother with 2 pitching positions in the first place?
Could be: usually the issue is a huge hole in front of the rubber, and they end up not completely inside the length of the rubber. I let that slide, but there's no prospect of ending up in the hybrid from doing that.
On what? You've asked about several dimensions of the pitching restrictions.

My preferred approach: "Knock it off!" That's it. They know exactly what I'm talking about, but if they want to make it a bigger problem, I'll address the bigger problem they make.
I like this too, though the explicit threat could just as easily remain implicit.
Too many words, and a bit too patronizing for me: the basis is correct, and they are indeed setting a poor example, but if the game is already hot then they're probably not going to be able to hear that.
Not true, unless I have (uncharacteristically) directed them to "go fight in the parking lot."
In general, it's fine to address inappropriate behaviors by telling people to stop doing them, but be careful offering a substitute (especially with adults).

So did I, and for the same reason: it's the most correct (and least incorrect) of the available responses.
It is not, as we know, the credited response.
Back in the day, I got paid (pretty well) to create questions for the GRE. I would not have been paid for that kind of question.

The standard term for this comes (IIRC) from J/R: tangle/untangle. It is indeed the same in all codes. Rich has posted the FED case play supporting the ruling.
I've never heard the term 'no call zone', nor do I care for its implication of an area where we make no calls. Tangle/untangle captures the idea better.

That's it.
Correct, and 9-1-1 NOTE 2 requires afford the defense an opportunity to appeal. Together, these provisions entail not leaving the field until the defense has a chance to appeal.
I'd treat the window for that opportunity as the same as any other half inning.

As this seems to be a 7-inning game, there's a good chance that FED rules apply.
The FED rule is slightly different (all runners must touch their advance base).
For the OP, when the BR fails to advance, this rule requires that the umpire not declare the game over after R3 scores. The reason is that the defense has the right to appeal the BR's failure to touch 1B.
And, though I'm not finding the rule at the moment, the window of opportunity for the defense to appeal will close when the infielders have all left fair territory. As an umpire, I'd stick around till then, and at that point depart with all due celerity.

The applicable rule requires such INT to be intentional. As described, I have R1 attempting to get to the base, and the throw pulls F6 into his path. If that's all that happened, the technical term is a 'train wreck', which denotes an unfortunate but incidental collision, with no violation by either player. In possession of the ball, F6 cannot be guilty of OBS; absent intent, R1 cannot be guilty of INT.
To rule INT here, we would need some clear and obvious action by R1 to go out of his way to hinder F6. Deviation from his path to the base, dropping a shoulder in a more aggressive move than merely protecting himself, reaching out to grab F6 or sweep his legs... that kind of thing.
Some umpires try to adjudicate this kind of play using the notion of "rights": R1 had a "right" to the base path or base or whatever, and F6 had a "right" to catch the ball or make a play or throw back to HP or whatever. Such attempts are generally misguided: for one thing, the rules seldom employ concepts of "rights."
But more important, if we are going to hold a player liable for violating some rule (INT, OBS, MC, etc.) each of those has its own criteria, and those criteria are not cashed out in terms of anyone's "rights." We properly apply the rule book criteria—here, intentional hindrance during a thrown ball—rather than get into a battle of pseudo-rights.

OBR does not use either term. FED's "followthrough" INT is what we are calling backswing INT, and FED treats it as a kind of batter INT (and applies the same penalty).
What FED calls 'backswing INT' isn't really anything: it's the batter bumping F2 before the pitch. Frankly, I'm not sure why we need a rule for that (as opposed to a mechanic).

Actually, I think Mr. Blue might be on to something. 6.03(a)(3) defines batter INT. OBR's concept of "backswing INT" is NOT batter INT (and not penalized as batter INT). It therefore requires its own criteria, which apparently, in OBR, do not involve hindrance, but only contact.
I had not noticed that before. It is misleading that we (not the rule) call it a kind of INT. Should be "backswing contact" or some such.

If that's all there is to the play, agreed.
If we stipulate that only F5 is protected, then I agree.
Not so fast. Once the ball gets past F5, as folks have been saying, protection transfers to F6. Don't confuse the provision in the rules about the batted ball going "through or by" a fielder, which excuses the runner behind him from interference with the ball, with the provision concerning interference with a fielder. The runner is not excused from that, and as described this play starts to sound like INT.

Rich, if I understand your question, it seems to be more of a mechanics question than a rules question. The rules question I take to have been (more or less) settled.
Again, if I understand, you seem to ask how we judge this play. Our positioning is not going to be very flexible: this is a batted ball, and this action will happen in a heartbeat. Moreover, it's complicated by F5 booting his end of play, so whatever quick expectations we might make when the ball is hit to his left will have to be revised on the fly.
But all the speed of the play does not require us to speed up. We need to see what happens and judge hindrance, bearing in mind that on a batted ball, runner/fielder contact is virtually always something, not nothing (2 esteemed members of my association notwithstanding).
It could be a call that requires 2 umpires to piece together what they see: PU has an angle on the runner/fielder interaction, and BU has an angle on what happens with the ball. If so, it's easier to get off a call made than to first make a call after conferring.

We need to call OBS whenever it happens. The key is to separate the infraction (which must be called) from the award (sometimes none).
Ordinarily, when we use the expression "the play stands," we're saying that there was no infraction, or that the infraction is ignored. That won't be the case with OBS. It would be more accurate in the case of OBS to say, "there is no further award," or the like.

I get the concern. I disagree with framing this as "penalizing a runner for a defensive miscue": the defense has a right to field the batted ball. That's the basis for conferring protection (from what? from liability for OBS) in the first place. If the runner hinders the defense's efforts to field the ball, then we penalize him for INT, not a defensive miscue.
If you think that's too high a burden on the runner, consider the burden on F5 who misplays the ball and loses the protection: if R2 bumps into him after his protection is gone, it's OBS, not INT. How many times have we heard coach complain, "so what, he's just supposed to disappear?" Well, yes. Is that fair? Yes: same rule for both teams.

If he's not out, you don't need the rule?
You need the definition of TAG, which requires that the ball be held securely in hand or glove, assuming the glove is properly worn. In pro rules, it's in the definitions section at the end. In HS rules this is 2-24-4: "A tag out is the put out of a runner, including the batter-runner, who is not in contact with his base when touched with a live ball, or with the glove or hand when the live ball is held securely therein by a fielder."
If his hand comes off, but the ball stays in the hand, that's also not an out.